An analysis of single stage axial-flow turbine performance using three-dimensional calculating methods.

Calhun: The NPS Institutinal Archive DSpace Repsitry Theses and Dissertatins Thesis and Dissertatin Cllectin 967-09 An analysis f single stage axial-flw turbine perfrmance using three-dimensinal calculating methds. Harrisn, Rbert Glen Mnterey, Califrnia. U.S. Naval Pstgraduate Schl http://hdl.handle.net/0945/635 Dwnladed frm NPS Archive: Calhun

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AN ANALYSIS OF SINGLE STAGE AXIAL -FLOW TURBINE PERFORMANCE USING THREE-DIMENSIONAL CALCULATING METHODS by Rbert Glen Harrisn Lieutenant, United States Navy B.M.E., University f Luisville, 959 Submitted in partial fulfillment f the requirements fr the degree f AERONAUTICAL ENGINEER frm the NAVAL POSTGRADUATE SCHOOL September 967

ABSTRACT The met ld f turbine perfrmance predictin develped by Vavra and Eckert has been refined in this analysis t realize mre f the ptential f the three-dimensinal calculating methds. Mach number and rtr tip clearance effects n blade utlet angles and lss cefficients have been lcalized rather than averaged ver the blade height. An apprximatin fr streamline curvature has been used. Perfrmance curves were determined fr tw single stage axialflw turbines lcated at the Prpulsin Labratry f the Naval Pstgraduate Schl. Test results were available fr ne f the turbines. Agreement between predicted and experimental perfrmance values was generally within 3 per cent.

' Thesis by Rbert G. Harrisn entitled: "An Analysis f Single- Stage Axial- Flw Turbine Perfrmance Using Three-Dimensinal Calculating Methds". ERRATA SHEET Page Line Change T 3 0 Develpement Develpment 3 2 ( ) (y/s) 3 22 58. if T tq /5l8.^ 8 3 develpement develpment 2 Eq.3 reads: = - V-f- = \^ *- a H - /2 2 Eq.lU reads: da = FZ dx a r 3 a mm m 2k 3 respresenting representing 26 7/8 calculat-ing calcula-ting 27 2 insert n a n after "the" a end f line. 27 27/28 prce-dur prce-dure 36 0 < X <V " 39 6 insert " " after "f" at end f line. kk 3/** pera- ting perat-ing k5 23 b unt blunt h9 8/9 ceffici-ents ceffi-cients 07 7 glade blade lhj Item 0 Item 0. shuld read: "This dcument is subje special exprt cntrls and each transmittal t freign natinals may be made nly with prir apprval f the Naval Pstgraduate Schl"

TABLE OF CONTENTS Sectin Page. Intrductin 7 2. Basis fr the Analysis 8 3. Tecbninue fr Obtaining a Slutin 23 A. MOD I and MOD II Turbines 37 5. Cnclusins and Pecmmendatins 48 6. Bibligraphy 88 Appendix A. Develpement f Equatins 89 B. Cmputatins f Outlet Angles and Lss 0 Cefficients C. Cmputer Prgram 2

LIST OF TABLES Table Page I. Sample Calculatins fr Statr Outlet 04 Angles II. Sample Calculatins fr Rtr Outlet 05 Angles III. Sample Calculatins fr Statr Lss 09 Cefficient IV. Sample Calculatins fr Stalling 0 Incidence and Rtr Lss Cefficients V. Sample Cmputer Output fr Statr 44 Slutin VI. Sample Cmputer Output fr Rtr 45 Slutin

. LIST OF ILLUSTRATIONS Figure. Turbine Thermdynamic Prcess 50 2. CrGinates and Streamlines 5 3. Velcity Diagrai! 52 4. Statr and Ptr Blade Prfiles (MOD I) 53 5. Statr and Rtr Blade Prfiles (HOD II) 54 6. Statr and Rtr Thrat Openings versus 55 Radial Psitin (MOP I & MOD II) 7. Variatin f Statr Outlet Angle with 56 Radius (MOD I) 8 Variatin f Statr Outlet Angle with 56 Mach Number (MOD I) 9. Variatin f Statr Outlet Angle with 57 dius 0 II) Variatin f Statr Outlet Angle with 57 lach Number (MOD II). Variatin f Ptr Outlet Angle with 53 Radius (MOD I) 2. Variatin f Rtr Outlet Angle with 53 Mach Number (MOD I) 3. Variatin f Ptr Outlet Angle with 59 Radius (MOD II) 4. Variatin f Rtr Outlet Angle with 59 I.ach Pumler (MOD ) 5. Statr Lss Cefficients as Functin f 60 Radius ("MOD I & MOD II) 6. Variatin f Blade Inlet and Stall Incidence 6 Angles with Radius (MOD I & MOD II) 7. Ptr Lss Cefficients as Functin f 62 Incidence Fati (MOD I) :-'tr Lss Cefficients as Functin f 63 Incidence Pati (MOD II)

Figure Page 9. Variatin f Referred Flwrate with 64 Referred RPM (MOD I) 20. Referred Mment versus Referred 65 RPM (MOD I) 2. Ttal-Static Efficiency versus Referred 66 RPM (MOD I, k=0.020 in.) 22. Ttal-Static Efficiency versus Referred 67 RPM (MOD I, k= 0.033 in.) 23. Ttal-Static Efficiency versus Isentrpic 68 Head Cefficient (MOD I) 24. Referred Pwer versus Referred RPM 69 (MOD ) 25. Theretical Degree f Reactin versus 70 Isentrpic Head Cefficient (MOD I) 26. Theretical. Degree f Reactin versus 7 Referred RPM (MOD I, P /P =.40) 27. Variatin f Referred Flwrate with 72 Referred RPM (MOD II, k=0.05 in.) 28. Variatin f Referred Flwrate with 73 Referred RPM (MOD II, k=0.033 in.) 29. Referred Mment versus Referred RPM L ' (MOD II, k=0.05 in.) 30. Referred Mment versus Referred RPM 75 (MOD II, k=0.033 in.) 3. Ttal-Static Efficiency versus Referred 76 RPM (MOD II, P /P =,30,.3) 32. Ttal-Static Efficiency versus Referred 77 RPM (MOD II, P /P =.40) 33. Ttal-Static Efficiency versus Referred 78 RPM (MOD II, P /P =.50,.5) 2 " 34. Ttal-Static Efficiency versus Referred 79 RPM (MOD II, P /P =.60) 35. Ttal-Static Efficiency versus Isentrpic 80 Head Cefficient (MOD II)

' ef : eferred. PM/\fe, k=0.05 >*. I OD 3. Etefei rt wer versus i - - 3?. -..i re< I ei vers us sri (MOD II, k=0 i 38. Theretic/ egree f eactin versus Isentrpic Head Cefficient (MOJ [I). 39. theretical Degree f Reactin versus ' ': II, P 2 40. Abslute Flw Outlet Angles as Functin f Radius (MOD II, k=0.05 in., P /P-^l.AO,. l/v =3,934) 4. Inferred Velcities as Functin f aclius (MOD II, k»0.05 in.,? /P-=.40, t RPM/^e" =3,934) / t'\ Relative Rtr Flw Outlet Angle as Functin 86 f Radius i. II, k»0.05 in., P /P *.40, =3,934) 43. Plts f Axial Velcity Patis versus Radius 87 Ratis at Peak Efficiency (MOD I & MOD II, P fv =.40) 2 44. Expansin frm Plenum 94 Cnditins at Exit f Blade Rw 9 ; "> 46..lade Gemetry 02 47. ip Clearance Factr X as Functin f Radius 03

TABLE OF SYMBOLS Symbls 2 A Area (in. ) a Thrat pening f blade channel (in.) B Shrud factr used in rtr lss cefficient calculatins (ditnensinless) b Blade's departure frm being straight-backed (in.) 2 2 C, Cnversin factr, 2gJ(ft -lb /sec -BTU) m c Blade Chrd (in.) c Specific heat, cnstant pressure (BTu/lb - R) E Kinetic energy rate (ft-lb/sec) e Mean radius f curvature f back f blade (in.) 2 g Universal gravitatinal cnstant (32.74 lb -ft/lb-sec ) H Ttal enthalpy (BTU/ lb ) H*** Bundary layer energv parameter ( >. ) h Static enthalpy (BTU/lb m ) h Blade height (in.) HP I Hrsepwer Integrand i Incidence angle (deg. r radians) Unit vectr i Stalling incidence angle (deg. r radians) J Cnversin factr (778.6 ft-lb/btu) j Distance frm threat t trailing edge f blade (in.) k Tip clearance (in.) k Isentrpic head cefficient (dimensinless)

Symbls L Distance between statins and and between statins and 2 (in.) M Mach number M Mment (ft- lb) m Mass flwrate (slugs/sec) m Expnent used in bundary layer calculatins, see Eq. 46 (dimensinless) P Pressure (psia) R Gas cnstant fr air (53.345 ft-lb/lb - R) m r Radius (in.) r* Theretical degree f reactin (dimensinless) s Blade spacing (in.) s Entrpy (BTU/lb - R) m s* Nn-dimensinal entrpy ( _J? ) c T Temperature ( R) t Blade thickness (in.) t Blade trailing edge thickness (in.) t* Prjectin f blade trailing edge thickness n the exit plane f the blade rw (in.) U Peripheral velcity (ft/sec) u Velcity within a bundary layer (ft/sec) V Abslute velcity (ft/sec) W Relative velcity (ft/sec) w Weight flwrate (lb /sec) m W. Fractin f the ttal flwrate which passes between the hub and any ther streamline (dimensinless) 2 W _ Reference flwrate (in. ref v ) ' 2

Symbls X Nn-dimensinal radius r where r is the mean,. m streamline radius r m X Shrud factr fr calculatin f rtr utlet angles (dimensinless) X See Eq. 45 e V V Y Nn-dimensinal axial velcity _A where A is the mean streamline axial velcity V m Y Pressure lss parameter, see En. 67 y Distance frm wall f a pint in a bundary layer (in.) Z Number f Mades Greek Letters C( Abslute pas flw angles (deg. r radians) ^O Gas flw angles relative t rtr (deg. r radians) /S Blade inlet angle (deg. r radians) # Specific heat rati (dimensinless) &g Streamline displacement, see Fig. 2 (in.) Bundary layer thickness (in.) SFeferred pressure P t (dimensinless) 477 f Streamline displacement, see Fig. 2 (in.) * Bundary layer displacement thickness (in.) f -k-k-k Bundary layer energy thickness (in.) * Lss cefficient (dimensinless) Efficiency (dimensinless) Nn-dimensinal distance frm the wall in a bundary layer ( ) Q Referred temperature 58. A (dimensinless) ^ Streamline curvature factr (dimensinless) 3

Greek Letters A Angle between flw and axis f turbine in a meridinal plane (deg. r radians) \ Factr used in predicting secndary lss cefficients, see Eq. 7 (dimensinless) ct Area restrictin factr (dimensinless) /O Density (lb /ft 3 ) / m / Nn-dimensinal flw functin CO Angular velcity (radians/ sec) Subscripts A Axial d Discharge E H Equivalent hub is Isentrpic expansin frm ttal inlet cnditins m Mean streamline Statin ahead f statr P P Prfile Pelative r Fadial ref re. Peferred Fenuired S Statr s Shrud s Isentrpic expansin frm enuivalent ttal cnditins T Tip t Ttal 4

Subscripts th u Theretical Tangential e Axial directin, cyclindrical crdinates Q Peripheral directin, cyclindrical crdinates Statin between statr and rtr 2 Statin after rtr Superscript ** Fefers t predicted values fr the mean streamline 5

. Intrductin Turbines frm an imprtant part f prpulsin systems. T ptimize a design it is necessary t knw the perfrmance at ff-design cnditins as well as the perfrmance at the design pint. Since the testing f a prttype is very cstly and time cnsuming, it is f great advantage t be able t predict the perfrmance characteristics by means f theretical methds. The mre advanced a system is the mre imprtant it becmes t imprve the accuracy f these methds. Hence it is necessary t base these methds n the fundamental laws f fluid dynamics rather than n ru.e-f -thumb apprximatins. The latter basis is pssible nly when data n previus designs are available. Fr new and advanced cnfiguratins, it will be necessary t apply refined methds which enable the designer and systems engineer t predict the effect f prpsed design changes. The principal equatins that describe the flw prperties in turbmachines are well knwn. These same equatins are used as a basis fr ail prper "three-dimensinal" calculating methds that have been cited in the technical literature. The methds differ hwever in the manner in which these equatins are manipulated and applied. This thesis is cncerned with the refinement f the threedimensinal methd f analysis develped by Vavra and Eckert. Based n the physical dimensins f particular test turbines that are available at the Turb-Prpulsin Labratry f the Naval Pstgraduate Schl, perfrmance curves were determined fr these machines. Cncurrently with the perfrmance analysis, experimental tests were cnducted n ne f these turbines s that actual experimental results culd be used t julge the accuracy f the prpsed theretical perfrmance evaluatin. In additin t his publicatins and classrm lectures which cnstituted the fundatin fr this analysis, Prfessr Vavra was very generus in prviding guidance and cunsel during the perid f this wrk. Fr this I am greatly appreciative, I wuld als like t thank Lieutenants P. M. Cmmns and J, A Messegee fr making their experimental results available. 7

2. Basis fr the Analysis The tw cnservatin equatins that must be satisfied t btain a slutin fr the flw in a turbine are the equatins f mtin and cntinuity. These equatins are satisfied at statins between blade rws. The methd used is that given by Vavra. Vavra develped the equatins f mtin and cntinuity fr abslute flws. The equatins in this frm are readily useable fr the psitin after the statr. Eckert later develped these equatins fr relative flws t 2 be used fr rtr calculatins. Eckert' s cnversin allws relative flw quantities t be handled directly withut cnversin t an abslute system. Eckert' s apprach als avids iteratin prcedures t determine the ttal enthalpy after the rtr. The assumptins made fr the develpement f the equatins used in the perfrmance analysis are:. An infinite number f blades in each rw s that dwnstream effects are nt felt upstream. 2. Axisymmetric flw at the statins where the equatins f mtin are slved. 3. Adiabatic and steady flw s that the ttal enthalpy alng any given streamline is cnstant thrugh the statr and the relative ttal enthalpy is cnstant thrugh the rtr. 4. All entrpy changes are assumed t ccur in the blade channels that are lcated ahead f the statins where the equatins f mtin are satisfied. Hence at the calculating statins the flw is assumed t be isentrpic alng particular streamlines. With the abve assumptins, the equatins f mtin fr abslute and relative flws, respecti/ely, are y - = \7 x (v*v^ + TVS () v H P = W x (7 x W + a uj) + T vs (2) vavra, M. H., Aer-Thermdynamics and Flw in Turbmachines. New Yrk, Lndn: Jhn Wiley and Sns, Inc.; 960, Chapter 6. 2 Eckert, R. H., Perfrmance Analysis and Initial Tests f a Transnic Turbine Test Rig (USNPGS Thesis, May 966), pp. 49-55. 8

W, V±tn Relative ttei enthalpy can be written as u, 2 U, 2 _ h \Nc Ul _ L. _ yl (3) Equivalent erthalv i«efined as Similar t ti.., the equivalent enthalpy H_ is cnstant alng a streamline fr the adpted assumptins. Equivalent enthalpy can als be written as -> 2. ^ H E =h^^ = K + + -r a (5) The intrductin f the equivalent enthalpy allws this quantity t be used fr the rtr in a manner analgus t the way ttal enthalpy is used fr the stetr. Fr equatins used in this analy* is, the subscripts refer t: - statin ahead f statr - statin ahead f rtr 2 - statin after rtr is - isentrpic expansin frm P s - isentrpic expansin frm P th - the theretical value E Figure is a temperature-entrpy diagram shwing the thermdynamic prcess alng a particular streamline fr a single stage turbine. In general, the fluid prperties will vary frm streamline t streamline, T>-e meth in which the lss cefficients are applied is als indicated in Fig.. The lss cefficients are defined a«jf s - V Fr the Statr (6) (. W* th -Wa Fr the Rtr (7) The crdinate system that will be used in the analysis is indicates in Fig. 2. This figure als shws the general layut f the ty >e turbine t which the predictin perfrmance analysis is applied. ier; cnventin fr the varius, angles that are needed in the analysts { indicated in /i&. 3. 9

. _ ' s The mdificatin f Eq. 2 int a frm that can be used fr the analysis is given in Appendix A, Sectin. The apprpriate frm f Eq. can be btained frm the mdificatin f Eq. 2 if the angular velcity a/ is set equal t zer. Other differences f the final equatins derived frm Eqs. and 2 are listed in Appendix A, Sectin. Equatin can then be written z ) \ As*l dt, - ~ <~,in~r< + -v! I jinck - x, Equatinn 2 becmes + Y,-VA^ dx, L Y, ~ a v^ - S M «, \ 'J dx, (8) dx VWY. W^Y^ %l^ dx^ Lw.iV, 2 " -jclxa where: y-^l r -^ fr Eqs. 8 and 9 respectively *** ^ (subscript m refers t mean streamline) (9) K=5.0 streamline curvature factr => r streamline displacements shwn in Fig. 2 L. + U L- a (see Fig. 2) c r The equatin f cntinuity is used in its nn-dimensinal frm by intrducing a flw functin <. Develpement f $ is given in Appendix A, Sectin 2. In differential frm the equatins fr abslute and relative flws can then be expressed by 4fe&fr-dA}«aA imtfiir) (0) and fferential element f area da is JA- I C c\r (2) 20

where: 2 number f blades a = blade exit pening (see Fig, 46) * arc. restrictin cefficient Since is valid fr isentrpic flw nly, the restrictin factr * must be intrduced t crrect the actual flw area t an effective area which accunts fr the restrictins due t the bundary layers n bth sides f the flw channel. 3 The factr ' can be expressed by In Eq. 3, * is the bundary layer displacement thickness and h*** is the s-called energy parameter defined as the energy thickness divided by the displacement thickness. The tern represents the lss that is assumed t ccur frm the inlet t the thrat f the blade channel, where ^J is the lss cefficient representing all the lsses acrss the rw f blades. The prfile lss 4 cefficient was used by Eckert t represent the lss prir t the blade thrat. Percentage f the ttal lss due t prfile lsses will vary cnsiderably depending n blade gemetry, radial psitin, and the incidence f the flw n the leading edge f the blade. Since secndary flw and tip clearance effects result in lsses in the blade channel, half the ttal lss cefficient prvides a better average representatin f the lsses in the blade channel prir t the thrat. Tie basis fr the develpement f C and H*** as used is given in Appendix A, Sectins 3 and 4. by multiplying and dividing by a^,, Eq, 2 is i/>- : - J.X : <. (4) Vavra, M, li,, Prblems f Fluid Mechanics in Kadial Turbmachines (lihde-saint-cenese, Belgium: Vn Karmin Institute fr Fluid Dynamics, 965) VKI Curse Nte 55b, pp. G46-50. *n i cckert, _p_, cit., p. 44. 2

: w After integratin Eqs. 0 and becme, respectively, and vvfr^ / r t 3 (5) (6) The flwrate w can be cmputed frm the cnditins ahead f the statr. Then a reference flwrate is defined by yve(t] t > J J where W is in square inches. Cntinuity will be satisfied fr the statr by (7) M -i dx STATOR = w ref Similarly fr the rtr 4 ^i dx»w rcf The in luence f the leakage flw thrugh the radial tip clearance (8) (9) has nt been accunted fr in Eq. 9. The element f area between the blade tips and the shrud is Afi= atrrdr (20) The flw thrugh the tip clearance area is -i Hi = 2. JX (2) "" "" J 7,P ip LEftP^NCE Since the tip clearance is relatively small, the values f I» w/ P, T, and X fr the tip will be used in Eq. 2. t E t E Z With Jith th( these assumptins Eq. 9 can be expressed by RcTdK vfyret L a w.*v«]ieerr The assumptins used t arrive at Eq. 22 are bviusly incrrect In tw respects. First, the flw represented by is nt 22

perpendicular t the tip clearance area. Secnd, the effective area represented by c is larger than that which prbably ccurs because f the relatively large bundary layers that exist n the shrud and blade tips. The exact behavir f the flw in the small regin between the shrud and the blade tips is impssible t predict withut further tests. Hwever, it is felt that the flwrate thrugh this space as represented in Eq. 22 is t large fr the reasns just mentined. Therefre a mre accurate apprximatin f this flwrate will be btained if the last term n the left side f Eq. 22 is divided by 2, yielding f Pf. tt lhx W(f(fHX) rip i!ij C+»»~ c' r*ji ROTOR The tip clearance flw included in Eq, 23 can be btained by integratin, Intrducing this expressin int Eq. 23 gives ROTO P. Wr l [aw2^] RarR (25) 3. Technique fr Obtaining Slutin With the equatins f mtin and cntinuity in the frms given by Eqs. 8, 9, 8 and 25, a methd has been develped t analyze single stage axial turbines, in particular, thse available fr test in the Turbine Te6t Rig f the Turb-Prpulsin Labratry f the Naval Pstgraduate Schl, The methd f analysis predicts turbine perfrmance fr specified values f inlet ttal pressure, inlet ttal temperature, rtr speed and the rati f ttal inlet t static discharge pressure p /P 2# This methd is similar t that described by Eckert. Hwever, Eckert 's analysis neglected sme effects which 5 Ibid, Sectin 3. 23

have been accunted fr in this develpetnent. Sme significant changes made in this methd are listed belw:. Statr and rtr utlet angles fr a particular radial lcatin are cmputed using a calculated Mach number fr that lcatin rather than an assumed Mach number r the Mach number f the mean streamline. 2. Variatin f lss cefficients due t changes in blade gemetry in the radial directin is accunted fr. 3. The influence f rtr tip clearance n the rtr utlet angles and lss cefficients is cncentrated near the tip f the blade rather than averaging these effects ver the full blade height. 4. Furth rder plynmials are used t better apprximate the curves respresenting blade characteristics as a functin f radius and the curves f rtr lss cefficients as a functin f incidence, 5. Streamline curvature effects have been accunted fr in the slutin f the equatins f mtin. In additin t the assumptins that were mentined in Sectin 2 fr the develpetnent f the particular frm f the equatin f mtin, the cnditins ahead f the statr are assumed t be unifrm; that is, the ttal temperature, velcity, and entrpy are assumed cnstant and the flw axial in directin. It is realized that cmpletely unifrm cnditins are difficult t btain, but any ther aesumpt n wuld be extremely difficult t develp mathematically. Direct slutin f the equatins f mtin is nt pssible since they are nnlinear in the dependent variable V, Likewise, n direct methd Is pssible t satisfy cntinuity. Slutins f these equatins must therefre be gained by making Initial assumptins fr the values f the axial velcities which must be Imprved by successive Iteratins until the equatins are satisfied. T accunt fr streamline curvature and slpe, a cmplete slutin f the flw thrugh the statr and rtr must first be made by neglecting the effects f curvature In rder t determine streamline lcetlns. Then the iteratin t accunt fr these effects may prgress. These requirements make the use f a highspeed cmputer a necessity. 26

This analysis has been prgrammed fr the IBM 360 cmputer using FORTRAN IV. The prgram is described in Appendix C. The fllwing paragraphs set frth the prcedural steps f the prgram. The equatins are listed in general frm withut referring t specific streamline lcatins. In the interest f clarity, hwever, sme relatinships will be written in frms similar t thse used in the prgram. Fr example, = f ^^V f [T i s (2)/Tf l*" wil re P re " sent the isentrpic relatinship fr the number 2 streamline. Five streamlines are utilized fr the analysis with the number streamline lcated at the hub and the number 5 streamline lcated at the tip as shwn in Fig. 2. The number 3 streamline will be used as the mean streamline, and the radius f this streamline will be referred t as the mean radius. The radial lcatins f the streamlines ahead f the statr will be such that the mass flwrate between adjacent streamlines is 25 per cent f the ttal flwrate. Psitins fr the streamlines after the statr and after the rtr are initially assumed. The lcatins f streamlines 2, 3, and 4 then vary during the slutin as necessary s that the percentage f the ttal flwrate between adjacent streamlines- des nt change. This cntinuity requirement will be called streamline cntinuity. Besides the radii, sufficient input infrmatin must be used t effectively reflect the physical characteristics f the statr and rtr blading. Sme f the physical prperties are intrduced directly; such as, the number f statr blades, the number f rtr blades, and the rtr tip clearance. The ther quantities used which reflect blade characteristics are thrat pening dimensins fr the blade channels, discharge angles, rtr blade inlet angles, lss cefficients, and stalling incidences fr the rtr. Thrat pening dimensin "a" is a functin f radius. The best methd fr intrducing this characteristic int the anaylsis is t enter the measured values f "a" tgether with the crrespnding radii. Then, utilizing the methd f least squares, a furth rder plynminal curve is fitted thrugh these pints. Frm the resulting plynmial, the value f "a" fr any radius required by streamline cntinuity can be determined. 25

. Discharge angles are nrndicted by using a cmbinatin f the methds f Vavra and Ainley. Outlet angles are first calculated using the frmula which Vavra established frm the experimental data f Beei. These angles are then crrected fr tip clearance, blade curvature, and Mach number effects with the methds given by Ainley. Statr and rtr discharge angles are predicted at three radii; namely, the hub, mean radius, and tip. The methd used fr calculating these angles is explained in Appendix B, Sectin. Values f statr gas utlet angles ex^ fr the mean streamline are determined tr Mach numbers M. f 0.5, 0.7, 0.75, 0.8, and.0. These values are represented by tw parablic curves f the frm a,= a+ bm,- cm* (26) The first curve is used fr Mach numbers M. frm 0.5 t 0.75 and the secnd established interim values f CX., fr M. between 0.75 and.0. Frm these curves the flw angle a, fr the mean streamline cnn be fund fr any Mach number M, The flw angle c, is als a junctin f radius r.. Therefre the changes f ex, ;r the hub and tip with reference t the mean radius, called A(X H and &&<? respectively, must be used. The flw angle at, <~an then be determined fr any r. by assuming a linear variatin t OC, between the hub and the mean radius and between the mean and the tip, With this assumptin and using the Mach number M. in Eq. 26 crrespnding t the number 2 streamline, the flw angle r, fr this streamline wuld be Ainley, D. c;, <md Mathiesn, G. C. R,, A Methd f Perfrmance estimatin fr Axial-Flw Turbines. Aernautical Research Cuncil, I & M N. 2974, 957. pp. 3-4. Beer, K, f Aerdynamic Design and Estimated Perfrmance f a Tw-Stage Curt is Turbine fr the Liquid Oxygen Turbpump f the M-l Engine. NASA CR 54764 (AGC 8800-2), Nv. 9, 965. p. 29. 26

The superscript ** is used with r- and flf in Eq. 27 t indicate the radius initially assumed fr the mean streamline and the cmputed fr that radius. Equatin 27 can then be used thrughut the analysis even thugh the radial lcatin f the mean streamline may change due t streamline cntinuity requirements. A similar apprach is used t establish the flw angle c> at the rtr discharge. The rtr blade inlet angles & are measured frm the manufacturing drawings f the blade prfiles. Using the values fp^ fr the hub, mean, and tip streamlines, a parablic curve is determined which gives J~) as a functin f X. Lss cefficients and stalling incidences are predicted by 8 using the methds f Ainley. Fr the present methd, the stalling incidence i is defined as that at which the lss cefficient is s twice the value f the minimum lss cefficient. Fllwing Ainley' s methds, lss cefficients are cmputed as a functin f the rati f flw incidence t stalling incidence -Jt-. Lss L 5 cefficients are als a functin f blade gemetry r radius. Since the statr has zer incidence, its lss cefficient is a functin f the radius nly. Statr and rtr lss cefficients and rtr stalling incidences are calculated fr the hub, mean, and tip radial lcatins. Ptr lss cefficients are determined fr values f yr ranging frm -2.0 t.6. Curves f g vs. ~ l are drawn fr each f the three radial lcatins, and values f &L, alng with the crrespnding nuantities -7-, are used t determine furth rder plynmials which apprximate these curves. A similar prcedur is fllwed t btain a furth rder plynmial representing stalling incidence i as a functin f radius f.. Sample calculatins fr the predictin f statr and rtr lss cefficients and stalling incidences are cntained in Appendix B, Sectin 2. Ainley, j>. cit., pp. 4-5. 27

Variatin f lss cefficients with radial lcatin is accunted fr by assuming a linear variatin f these quantities between the hub and the mean radius and between the mean and the tip. T demnstrate the prcedure fllwed in cmputing rtr lss cefficients, the fllwing example is given. Fr a particular incidence, the first step in the determinatin f ^f fr the number 2 streamline wuld be t calculate i by using the radius r, f that streamline and the plynmial f the frm i i (r.). Lss cefficients fr the hub and mean radius wuld then be cmputed using the resulting -4 in the plynmials, fr these radii, f the frm % = <7C (-^-). With the assumed linear variatin J (z\ wuld be %w-%m r**-r,( 3;- x (28) The reasn fr using the superscript ** n the mean streamline value* Is the same at previusly mentined in cnnectin with Be. 27. Fr the first apprximatin, the Mach number M ahead f the statr is assumed, and the static prperties and flwrate at statin are rund by T e < v frgr-r; M (>9) (30) '"a (3) /: (3?) (33) vv«/4^v (34) The reference flwrate W statins i and 2. f will ba used t check cntinuity at 2H

The next step is t determine axial velcities after the statr that satisfy the equatin f mtin. Ttal enthalpy after the statr is cnstant by assumptin, and streamline curvature effects are neglected at this stage f the analysis. With these cnditins, Eq. 8 simplifies t This equatin can be integrated t give X (36) ** / t where X is arbitrary and *- //?C s the cnstant f integratin. Using the bundary cnditin Y.0 at X».0, /t)c can be fund then by 0=fUt + lhc t n /ac z =-J~JcJX (37) Eauatin 38 must be expressed in a frm that can be utilized in the cmputer. Expansin by infinite series yields ^ (38) The quantities cntained in I, Eq. 36, must be evaluated befre prceeding with the slutin. Fr the first apprximatin, assumptins are made fr the values f Y, M, and V.. The flw m angles (%. are then calculated by using Eqs. 26 and 27. After the value f C[. has been determined fr each streamline,,^7 I is arx i cmputed. This derivative and all thers needed in the analysis are fund by finite difference methds. Enthalpy is cmputed by (39) 9 and the entrpy term is fund by the methd f Vavra. (40) 9 Vavra, M. H., Aer-Thermdynamics and Flw in Turbmachines. New Yrk, Lndn: Jhn Wiley and Sns, Inc., 960. pp. 445-447. 29

. ] JL, I - Y* V^ (4) t Eq. 28. Each time a slutin t Eq. 36 is fund, the new values f Y are then used fr the next iteratin. After five iteratins, 0( and -rr are recalculated using the new statr exit Mach numbers, The Mach number fr each streamline is fund by (42) V.- csc. (42a) 'l 't agjcp (A3) With the new values fr dc, ( and five mre iteratins are made t determine the crrected values f Y. The quantities represented by Eqs. 42-44 are recmputed after satisfying the equatin f mtin and additinal quantities determined by T* % * \ (44) (45) (46) The flwrate thrugh the statr is cmputed next and cmpared with that required t satisfy cntinuity. Reference flwrate between the hub and each streamline is fund frm / / (47) (48) where H»[(i) -(tp] (49) 30

<*b O The "a" and "a " in bq. 4b are fund frm the plynmial which represents thrat pening as a functin f radius. re. is slved, the pressure rati is cmpared with the critical pressure rati, If the critical pressure rati has been exceeded, the flw is chked at that radial lcatin and the critical pressure rati is used in calculating ±> and ~ The fractin f the ttal flwrate passing between the hub and each streamline is cmputed by y y wi.lx Ttal flwrate, as fund by the denminatr f Eq a 50 multiplied hy -... r»'... is then cmpared with the reference flwrate. Overall cntinuity is satisfied if the difference is less than 0.0002, If required and cmputed flwrates are nt within this tlerance, the axial velcity fr the mean streamline is adjusted by ; f^~ ' W;-^ j; " i V " -v,, *«. (N.V) VA I wl L O.OOQ6? is the required reference flwrate divided by a^.h^v^, and (5), is the denminatr f Eq, 50, cmputed Slutins t the equatin f mtin and cntinuity are successively fund until verall cntinuity is satisfied, The fractins f the ttal flwrate determined fr each streamline by Eq, 50 are then cmpared with the crrespnding flwrate fractins ahead f the statr, Streamline cntinuity is satisfied if agreement is within 0.002 fr each streamline. If stream* line cntinuity has nt been satisfied, the streamlines in errr are adjusted by = X OLD +fw f -w f, ':.'/ '^OLO Tree,. T cri il \ J JW, (52 > equatin 52 applies nly t stieamlines 2, 3, and 4 since W = at the hub and W ~.0 at the tip. 3

I cs^',. If streamline psitins have been adjusted, new streamline radii are fund by NEW ~ X(s) W Cu 0LO (53) and new values f X fr each streamline are btained by X=7^ (54) ^ NEW With the new streamline radii the equatin f mtin and verall cntinuity must is be satisfied again. The fllwing values are determined after streamline cntinuity satisfied: U f "(30)(I2) > (55) U,= {f- U, (56) I'Vu.-U, (57) / Wu, W (59) T te =T, ^ -j- + 2-3 Cp EgTE; <60) H E =Tt t c P (6i) /T te N*^" Pt E =P) T/ (62) Rtr blade inlet angles are determined fr each streamline by means f the parabla which establishes pq as a functin f X. Incidence is fund by = ^,-/4 ^ 32

~ btalling incidences and rtr lss cefficients are then determined using the prcdure described in cnnectin with Eq. 28. If the incidence rati -4 is less than -2.0 r greater than.6, ; 's the value fr -4 is set equal t -2.0 r.6, respectively. One f the quantities necessary fr the equatin f mtin after the rtr is - ba. This quantity is separated int tw parts, d r^z -'L^'9.. represents the entrpy gradient due t changes f entrpy acrss the statr and referred t statin 2. The term d *a represents the entrpy gradient due t the different entrpy changes thrugh the rtr. The entrpy increase thrugh the rtr is cmputed by using the crrespnding values f the rtr in Eq. 4; fr examplejh wuld be used instead f H. Neglecting streamline curvature and slpe, Eq. 9 can be rewritten The discharge angle /^ and its derivative -=p-* are fund in the same manner as previusly described fr the statr discharge angles. The methd fr slving Eq. 65 is similar t that used fr Eq. 36 with the exceptin that iteratins are carried ut until crrespnding values f Y change by less than 0.005 y r un>.il 3 iteratins have been cmpleted. The extra iteratins are necessary because there may be a slwer cnvergence f the values f Y at statin 2. After satisfying the equatin f mtin, values at statin 2 crrespnding t thse represented by Eqs. 42-47, are cmputed. It shuld be nted that where abslute velcity terms are used in Eqs. 42-47, the crrespnding equatins fr the rtr will emply relative velcities. Therefre M is the Mach number f the flw 2 relative t the rtating rtr blade. 33

Overall cntinuity is checked at the rtr discharge using the same prcedure utilized fr the statr; hwever, the reference flwrate is increased accrding t Eq. 25 t accunt fr the flwrate between the blade tips and the surrunding shrud. After verall cntinuity has been satisfied at statin 2, streamline cntinuity is checked. In additin t the functins perfrmed fr the check after the statr, there are certain quantities that must be adjusted fr streamline relcatin. These are ' ^ "'/new \^^ 2 /0Lt. <:IX 2 (66) (67) (*fl w '(^"JL. + TtJ 2, C x *M (68) A slutin exists when the equatins f mtin,and verall and streamline cntinuity have been satisfied; hwever, this slutin has neglected streamline curvature and streamline slpe. T btain a slutin which accunts fr streamline curvature effects, the terms in Eqs. 8 and 9 that cntain %r r LR must be included when slving the equatins f mtin. The streamline displacements >' and DR, shwn in Fig. 2, are cmputed fr each streamline by and Sr=r, - k ^ (70) AR-r.-r, (7i) Equatin 70 Is based n the assumptin that the radii f any streamlint; at statins and 2 are nt greatly different. Therefre the csine f the angle between the line cnnecting these pints and the axis f the machine is apprximately equal t unity. The length L, als shwn In Fig. 2, is taken t be half the distance frm 0. inch ahead f the statr t 0. inch after the rtr. 34

After a slutin is btained which accunts fr streamline P curvature effects, the resultant pressure rati =2 is P a P* t cmpared with the - specified initially. Ideally fr each streamline will be the same. Hwever, cmputer slutins fr this quantity may vary slightly frm streamline t streamline. Therefre a mass-flw-weighted value f ~, called ( ^ ), is fund by Z 3- (72) C-l If the specified ^- and ( -). differ by mre than 0.0003, the Mach number f the flw ahead f the statr is prperly adjusted and anther slutin is fund. P After Che iteratins fr i^ have been cmpleted, additinal P& quantities are determined fr each streamline, using AH = H,-H --(u V,-U,V i )gy (73) Overall efficiency is then cmputed by f" Alu.-Tfc-T.fr (76) The ideal change in enthalpy Ahi^ is the isentrpic enthalpy drp frm the ttal inlet pressure F\ t the static discharge pressure F. Equatin 76 is used t cmpute the efficiency f a 35

single stage turbine because the kinetic energy leaving the rtr cannt be utilized. The efficiency defined by En. 76 will be referred t as ttal-static efficiency. Theretical degree f reactin r* and head cefficient k. are is given by and r = (77) Fr turbine perfrmance curves it is desirable t btain the mass-flw-weighted values f efficiency ( )., w head cefficient (k. )., is'w' theretical degree & f reactin (r*)., v 'w' hrsepwer v (HP)., v 7 w' and The last tw nuantities are fund frm and <V*- (78) (alilih '"^'W " SSO (79) M* - ^ ^ The mass-flw-weightedzla/, called ( H)., as well as ( 7? )., (k. )., and (r*). are cmputed using enuatins similar t E. 72. is w w r Referred values are btained, fllwing NASA practice. Fr -.4: (80) ///* = ( HP )*t (8) f fe S M<> rer = (J^U^ (82) WM+f = -4^- (83) > VVVe/ = W SW (84) x/ V (84a) 36

where: g- ft _ T t_ Tst. ~ 3/8.4 (85) ~ Psr. " M.7 (86) A. MOD I and MOD II Turbines The methd f analysis as presented was used t determine perfrmance curves fr the s-called MOD I and MOD II turbines. Bth turbines are single stage axial-flw machines. Experimental tests were cnducted n the MOD II turbine by Cmmns and Messegee and are described in Fefs. 4 and 6. The test results are pltted with, apprpriate predicted perfrmance curves. The s-called MOD I turbine was designed fr free-vrtex flw and has highly twisted blades. Outer diameter f this turbine is 9.898 inches. The hub diameters f the statr inlet and rtr discharge are 6.930 and 5.970 inches, respectively. The statr cntains 3 blades and the rtr 22 blades. Blades f the MOD I turbine are generally thin. The statr and rtr prfiles used t predict utlet angles and lss cefficients are shwn in Fig. 4. The MOD II turbine is apprximately the same size as the MOD I, but its blading is cnsiderably different. The blades f the MOD II turbine are thick with blunt leading edges and cnstant prfiles ver the blade height. Outer diameters f the MOD II turbine statr and rtr are 9.70 and 9.836 inches, respectively. The statr has a hub diameter f 6.796 inches, and the hub diameter f the rtr is 6.598 inches. There are 9 statr blades and 8 rtr blades. Statr and rtr blade prfiles fr the MOD II turbine are shwn in Fig. 5. Thrughut the remainder f this thesis the MOD I and MOD II turbines will be referred t simply as MOD I and MOD II. The minimum thrat pening "a" f the blade channels is a very critical dimensin. Slight variatins f this quantity have a cnsiderable effect n turbine perfrmance. Since this quantity is s sensitive, values f "a" measured frm the actual hardware were used fr the analysis rather than thse btained frm the manufacturing 37

( drawings. Then errrs due t manufacturing will nt be a factr in cmparisn f predicted and experimental results. Figure 6 shws the thrat penings "a" as a functin f radius fr the statr and rtr blading f bth turbines. Predicted statr utlet angle 0(^ as a functin f radius r- fr the MOD I is pltted in Fig. 7. The assumed linear variatin f X. between the hub and the mean radius and between the mean and the tip is readily apparent in this figure. Figure 8 shws the predicted variatins f \ ( with Mach number M fr the MOD I. Figures 9 and 0 are the crrespnding plts fr the MOD II. Relative discharge angles j8z were cmputed fr tw radial tip clearances k fr each turbine; namely, 0.020 and 0.033 inches fr the MOD I, and 0.05 and 0.033 inches fr the MOD II. The predicted flw angles z. are pltted in the same manner as previusly described fr the flw angles c\. Figures and 2 shw / _ as a functin f radius r- and as a functin f Mach number M-, respectively, fr the MOD I, where M_ refers t the Mach number f the flw relative t the rtr. Figures 3 and 4 shw the crrespnding plts fr the MOD II. The predicted effect f radial tip clearance n the discharge angles p z can be seen in Figs. - 4. Statr lss cefficients li were cmputed fr the radial lcatins crrespnding t the hub, mean radius, and tip. Figure 5 shws 'ff as a functin f radius r fr bth the MOD I and MOD II. The straight lines in this figure between the values f at the hub and mean radius^and between the mean and the tip, reflect the assumed linear variatin f ^ c in these regins. Variatin f rtr blade inlet angle A., with radius r is pltted fr bth turbines in Fig. 6. The difference between the untwisted and the free-vrtex blades is easily nted in this plt. Als shwn in this figure are curves representing the variatin f stalling incidence i with r... The change f the MOD I rtr blade prfiles with radius is reflected by a cnsiderable variatin f i whereas just the ppsite is true fr the MOD II. 38

Curves fr the MOD I shwing predicted rtr lss cefficients Uf as a functin f incidence rati -4 fr the hub, mean radius, and tip are shwn in Fig. 7. Since the lss cefficients between th( mean radius and the tip are dependent n tip clearance, there are tw curves fr the tip. One curve hlds fr the tip clearance f 0.020 inches; the ther is fr the larger tip clearance f 0.033 inches. Fr negative incidence ratis between -0.5 and -2.0 the curves fr the tip are estimatins. This was necessary because the cmputatins by the methd shwn in Appendix B, Sectin 2, gave unrealistically lw values f ^L as the flw inlet angle A apprached -90. The situatin is mre easily understd when it is nted that the blade inlet angle is -33.7 and the stalling incidence is 37 at the rtr blade tip. Figure 7 shws that the lss cefficients fr the hub (r=3.300 in.) are relatively large. The lss cefficients at the hub are larger fr mst incidence ratis than thse at the tip fr a tip clearance f 0.020 inches. The larger value f at the hub reflects the higher lsses that are assciated with an impulse type blade. It can be seen in Fig. 4 that the blade shape varies frm a reactin type prfile at the tip t an impulse type prfile at the hub. Lss cefficients fr the MOD II rtr are pltted in Fig. 8 fr tip clearances f 0.05 and 0.033 inches. The predicted similarity f the curves fr the hub and mean radius is t be expected since the blading differs nly in slidity. Althugh the blade prfile is the same at all radii, the lsses due t tip clearance result in larger lss cefficients fr the tip prfile. Fr cnvenience, the blade prperties used fr calculating the MOD II rtr lss cefficients were thse at the hub, mean and tip radii f the rtr discharge. Since the annulus area at the rtr discharge is larger than the annulus area at the statr discharge and since the flw incidence is a significant parameter fr the rtr lss cefficients, it wuld have been mre apprpriate t use the blade characteristics at the hub, mean, and tip radii f the rtr inlet. Hwever, the errr is insignificant because the blade prfile des nt change alng the radius. 39

It may be nted frm Fig. 4 that the minimum radius, fr which a blade prfile is given, is 3.597 inches whereas the radius at the hub f the MOD I statr ex^t is 3.300 inches. The utlet angle and lss cefficient fr the hub were fund by extraplatin, using the values cmputed fr the radii f 4.25 and 3.597 inches and assuming a linear variatin f these quantities with radius. The relative utlet angle fr the hub f the MOD I rtr was fund in a similar manner. Perfrmance curves fr the tw turbines were determined fr the rtr tip clearances mentined earlier. The ttal inle t t static discharge pressure ratis investigated were.30,.40,.50, and.60, with the exceptin that pressure ratis f.3 and.5 were used fr the MOD II with 0.033 inch rtr tip clearance. These pressure ratis agree mre clsely with thse experimentally investigated by Cmmns and Messegee. The exial distances L used fr the determinatin f the curvatures depend n the axial clearance between the statr and rtr as well as n the blade gemetries. Axial clearances f 0.4 and.0 inches were used fr the analyses f the MOD I and MOD II, respectively, Curves representing the perfrmance f the MOD I are pltted in Figs. 9 thrugh 26. Perfrmance values pltted are mass-flwweighted values unless stated therwise. Figure 9 shws referred flwrate as a functin f referred RPM. The increase in flwrate due t an increase in rtr tip clearance can be seen in this figure. Althugh the flwrate is greater fr the larger tip clearance, the ti que develped is greater at the smaller tip clearance. The decrease in trque fr the larger tip clearance results frm the increased lsses and decrease in turning angle f the flw thrugh the rtr near the tip. The predicted effect f tip clearance n trque is shwn in Fig, 20 where referred mment is pltted versus referred RPM. The variatin f ttal-static efficiency with referred RPM fr the tw tip clearances can be seen in Figs. 2 and 22. The referred RPM at which maximum efficiency ccurs increases when the ttal inlet t static discharge pressure rati is increased. Blade lsses 40

p. and the kinetic energy f the flw leaving the rtr affect the ttalstatic efficiency. As pressure rati is increased the abslute velcity leaving the statr and the relative velcity leaving the rtr increase. Therefre the peripheral speed f the rtr must increase t btain cnditins where the abslute velcity leaving the rtr is in an axial directin and where the relative flw ahead f the rtr has zer incidence. The RPM where the flw has zer incidence n the rtr will nt necessarily be that at which the abslute velcity leaving the rtr is axial. At any RPM, flw incidence and abslute discharge angles vary frm streamline t streamline, and the abve statements refer t mass -flw-weigh ted values. It may be nted in Figs. 2 and 22 that the peak ttal-static efficiency decreases smewhat as the pressure rati increases. At higher pressure ratis the rati f kinetic energy leaving the rtr t the wrk dne n the rtr increases. Since the kinetic energy leaving the rtr is lst energy fr a single stage turbine, the ttal-static efficiency declines. The effects f pressure rati and tip clearance n efficiency can be seen in Fig. 23 where ttal-static efficiency is pltted as a functin f the isentrpic head cefficient. The variatin f referred pwer with referred RPM can be seen in Fig. 24. Peak pwer des nt ccur at the same referred RPM at which peak efficiency ccurs. The peak referred pwer ccurs at the referred RPM where the prduct f ttal-static efficiency and referred flwrate is greatest. Theretical degree f reactin is pltted as a functin f isentrpic head cefficient in Fig. 25. It may be nted that the theretical degree f reactin increases with increasing pressure rati and decreases with increasing tip clearance fr any given isentrpic head cefficient. The predicted effects are cnsiderably different frm the results f the radial turbine tests cnducted by Riley. Riley fund that theretical degree f reactin was independent f pressure rati and axial clearance fr radial turbines. 0 Riley, M. W., The Effect f Axial Clearance n the Perfrmance f a Dual Discharge Radial Turbine (USNPG Thesis, December 966^ 70., 4

Perfrmance values fr each streamline are btained frm the cmputer slutin. Hwever, plts utilizing values fr each streamline wuld be difficult t analyze. The deviatin f the hub and tip values frm that f the mass- flw-weighted average may be seen in Fl. 26. In this figure, hub, tip, and mass-flw-weighted values f theretical degree f reactin are pltted as functins f referred RPM fr a ttal inlet t static discharge pressure rati f.40. Perfrmance curves fr the MOD II crrespnding t thse presented fr the MOD I are pltted in Figs. 27 thrugh 39. Additinal plts have been used fr the MOD II because f the inclusin f experimental results. An axial clearance f.0 inches was used fr the theretical predictin. Therefre, nly experimental data fr that axial clearance are shwn. Cmments made cncerning the perfrmance curves f the MOD I are applicable t the MOD II perfrmance curves als. Differences in the perfrmance f the tw turbines will be discussed later. Plts f the variatin f referred flwrate with referred RPM are shwn in Figs. 27 and 28 fr tw rtr tip clearances. The quantitative values as well as the curve shapes agree well with the experimental data. The maximum difference between predicted and experimental referred flwrates ccurs at a pressure rati f.5 fr a tip clearance f 0.033 inches. There are tw experimental pints fr this pressure rati and tip clearance that differ frm the predicted curve by abut 2 per cent. The predicted and experimental values fr all pressure ratis and tip clearances have an average difference f less than per cent. Figures 29 and 30 shw curves f referred mment versus referred RPM. The trends expressed by the predicted curves ere in excellent agreement with the experimental data. Althugh the quantitative agreement between theretical and experimental values is very gd fr three f the curves, the experimental trque is generally lwer than the predicted trque. The average difference between predicted and experimental values is abut 3 per cent. H

Figures 3 thrugh 34 shw ttal-static efficiency as a functin f referred RPM. The shapes f the predicted curves generally agree well with the experimental results. Hwever, there is an indicatin that experimental efficiencies decrease mre rapidly at high RPM than is predicted by the theretical curves. In the high RPM regin the upper part f the rtr blade has a large negative flw incidence. In the predictin analysis when the incidence rati had a value less than -2.0, the value f -2.0 was used fr cmputing lss cefficients. This limitatin may be the reasn that the predicted efficiencies in the high RPM range d nt decrease as rapidly as the test data indicate. The quantitative agreement between predicted and experimental efficiencies varies cnsiderably between different pressure ratis and tip clearances. There are tw data pints at a pressure rati f.3 and tip clearance f 0.033 inches where the experimental efficiencies are ver five pints belw the predicted values. At a pressure rati f.50 and a tip clearance f 0.05 inches the average difference between predicted and experimental efficiencies is.5 pints. Giving equal weight t all experimental values the average difference between experimental and predicted values is 2.6 pints. The calculatins gave a decrease in efficiency by abut tw pints as the tip clearance was increased frm 0.05 t 0.033 inches. The decrease in experimental efficiencies fr the increased tip clearance varied with the different pressure ratis. Hwever, the average decrease in efficiency is clse t the predicted decrease. Ttal-static efficiency as a functin f isentrpic head cefficient k. is shwn in Fig. 35 fr pressure ratis f.40 and.60. Figure 38 shws degree f reactin r* versus k. N experimental data are pltted in these figures because experimental values f mass-flw-weighted r* and k were nt available. Plts f referred pwer as a functin f referred RPM are shwn in Figs. 36 and 37. The cmments made earlier cncerning the referred mment plts apply t these curves als, since the turbine pwer is prprtinal t the prduct f trque and RPM. ^3